| L(s) = 1 | + (1.20 − 0.777i)2-s + (−0.238 + 1.66i)3-s + (−0.802 + 1.75i)4-s + (−2.65 − 9.04i)5-s + (1.00 + 2.19i)6-s + (−5.31 + 4.60i)7-s + (1.21 + 8.44i)8-s + (5.93 + 1.74i)9-s + (−10.2 − 8.87i)10-s + (1.86 − 2.90i)11-s + (−2.72 − 1.75i)12-s + (7.08 − 8.18i)13-s + (−2.84 + 9.70i)14-s + (15.6 − 2.24i)15-s + (2.97 + 3.43i)16-s + (−8.16 + 3.72i)17-s + ⋯ |
| L(s) = 1 | + (0.604 − 0.388i)2-s + (−0.0795 + 0.553i)3-s + (−0.200 + 0.439i)4-s + (−0.531 − 1.80i)5-s + (0.167 + 0.365i)6-s + (−0.759 + 0.657i)7-s + (0.151 + 1.05i)8-s + (0.659 + 0.193i)9-s + (−1.02 − 0.887i)10-s + (0.169 − 0.263i)11-s + (−0.227 − 0.145i)12-s + (0.545 − 0.629i)13-s + (−0.203 + 0.693i)14-s + (1.04 − 0.149i)15-s + (0.185 + 0.214i)16-s + (−0.480 + 0.219i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02910 - 0.0777361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02910 - 0.0777361i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-22.4 - 4.93i)T \) |
| good | 2 | \( 1 + (-1.20 + 0.777i)T + (1.66 - 3.63i)T^{2} \) |
| 3 | \( 1 + (0.238 - 1.66i)T + (-8.63 - 2.53i)T^{2} \) |
| 5 | \( 1 + (2.65 + 9.04i)T + (-21.0 + 13.5i)T^{2} \) |
| 7 | \( 1 + (5.31 - 4.60i)T + (6.97 - 48.5i)T^{2} \) |
| 11 | \( 1 + (-1.86 + 2.90i)T + (-50.2 - 110. i)T^{2} \) |
| 13 | \( 1 + (-7.08 + 8.18i)T + (-24.0 - 167. i)T^{2} \) |
| 17 | \( 1 + (8.16 - 3.72i)T + (189. - 218. i)T^{2} \) |
| 19 | \( 1 + (3.35 + 1.53i)T + (236. + 272. i)T^{2} \) |
| 29 | \( 1 + (10.2 + 22.4i)T + (-550. + 635. i)T^{2} \) |
| 31 | \( 1 + (1.70 + 11.8i)T + (-922. + 270. i)T^{2} \) |
| 37 | \( 1 + (3.28 - 11.2i)T + (-1.15e3 - 740. i)T^{2} \) |
| 41 | \( 1 + (-25.6 + 7.54i)T + (1.41e3 - 908. i)T^{2} \) |
| 43 | \( 1 + (16.9 + 2.44i)T + (1.77e3 + 520. i)T^{2} \) |
| 47 | \( 1 + 40.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + (12.2 - 10.6i)T + (399. - 2.78e3i)T^{2} \) |
| 59 | \( 1 + (-37.3 + 43.0i)T + (-495. - 3.44e3i)T^{2} \) |
| 61 | \( 1 + (-100. + 14.5i)T + (3.57e3 - 1.04e3i)T^{2} \) |
| 67 | \( 1 + (-64.9 - 101. i)T + (-1.86e3 + 4.08e3i)T^{2} \) |
| 71 | \( 1 + (-79.3 + 50.9i)T + (2.09e3 - 4.58e3i)T^{2} \) |
| 73 | \( 1 + (-4.06 + 8.89i)T + (-3.48e3 - 4.02e3i)T^{2} \) |
| 79 | \( 1 + (63.5 + 55.0i)T + (888. + 6.17e3i)T^{2} \) |
| 83 | \( 1 + (7.26 - 24.7i)T + (-5.79e3 - 3.72e3i)T^{2} \) |
| 89 | \( 1 + (53.1 + 7.63i)T + (7.60e3 + 2.23e3i)T^{2} \) |
| 97 | \( 1 + (5.80 + 19.7i)T + (-7.91e3 + 5.08e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.27841163781643898500807201751, −16.26013842630526219164371168931, −15.44965873046576662875930630313, −13.10229271140146896930675743647, −12.82142817576058278949916828538, −11.45619937184023491734597617777, −9.341243657512377384185349602664, −8.264486604279009806358102101927, −5.23852155342581675172634755309, −3.93490285097925138220162514775,
3.82868077097028001470722994646, 6.65592999625005866327429411815, 6.98324800515773236837257844223, 9.890854185767907591882634875005, 11.07597662770785211394998317904, 12.91441550770227919566723078528, 14.02371930051774131368399299284, 15.00786521742152882701929632541, 16.04908408318019978721523954709, 18.08574853755615813828213365520
